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Re: ds^2 and metric tensor

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On Saturday, November 19, 2016 at 1:04:52 AM UTC-8, Maciej Woźniak wrote:
> Użytkownik "JanPB"  napisał w wiadomości grup 
> dyskusyjnych:42b3857f-9843-4954-afa3-964e45541b86@googlegroups.com...
> 
> On Friday, November 18, 2016 at 5:12:38 AM UTC-8, mlwo...@wp.pl wrote:
> > A question for both these mumbling halfbrains
> > pretending gurus here:
> > So , ds^2 in the formula ds^2= dx^2/x^4 + dy^2
> > is a metric tensor, right?
> > If it is, dx^2/x^4 +dy^2 must be a formula giving
> > a real number scalar. Is it really?
> 
> |It gives a real number if you plug in a pair of tangent vectors (say, u and 
> v)
> |at a point (say (x,y)) into ds^2. Let's do that, for example consider these
> |u and v:
> 
> |  ds^2|_(x,y)(u,v) = (1*3)/(4^4) + 2*(-7) = 3/256 - 14 = -3581/256
> 
> 
> So,  poor idiot, we have:
> 1)ds^2
> 2)ds^2|_(x,y)(u,v)
> You're a true idiot, but I think you can still notice some difference,
> can't you?

Is it possible for you to talk like an adult?

> Which one gives a real number? The second one, right?

Right.

> And the second one  is your tensor.

No, the first.

> 1) is a square of differential.

No. It's not a square of anything. See also the comment on the term "differential" below.

> Of course, you don't know what
> it means, but I write for others, not for you.

That's fine, but note that the word "differential" is not well-defined in mathematics
UNLESS, typically: either (1) one is doing Abraham's non-standard analysis, or (2) one uses
the word "differential" to denote the exterior derivative of a function, "df", which is a covector,
or some combination of such.

Of course in applications one always knows how to interpret the word "differential"
(it always refers to some implicit limiting process, typically left unsaid and presumed obvious
from the context - this is where "mathematical maturity" comes in, you've probably
heard the term). Given your extraordinarily dissipated "concept map" we cannot really
go this route here.

So if you use the word "differential", please define _precisely_ what you mean by this.
Otherwise we're discussing poetry (nothing wrong with that, just let us know beforehand
that this is what you're interested in).

Bottom line is the tensor ds^2 is not a square.

--
Jan

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ds^2 and metric tensor mlwozniak@wp.pl - 2016-11-18 05:12 -0800
  Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-18 12:26 -0800
    Re: ds^2 and metric tensor Hank Barco <hankbarco@hankbarco.org> - 2016-11-18 21:05 +0000
    Re: ds^2 and metric tensor Maciej Woźniak <mlwozniak@wp.pl> - 2016-11-19 10:04 +0100
      Re: ds^2 and metric tensor paparios <paparios@gmail.com> - 2016-11-19 05:56 -0800
        Re: ds^2 and metric tensor Maciej Woźniak <mlwozniak@wp.pl> - 2016-11-19 15:24 +0100
        Re: ds^2 and metric tensor Gary Harnagel <hitlong@yahoo.com> - 2016-11-19 06:30 -0800
      Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-19 15:55 -0800
        Re: ds^2 and metric tensor Maciej Woźniak <mlwozniak@wp.pl> - 2016-11-20 08:49 +0100
          Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-20 01:35 -0800
            Re: ds^2 and metric tensor Maciej Woźniak <mlwozniak@wp.pl> - 2016-11-20 14:17 +0100
              Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-20 20:38 -0800
                Re: ds^2 and metric tensor mlwozniak@wp.pl - 2016-11-20 22:54 -0800
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-20 23:08 -0800
                Re: ds^2 and metric tensor Maciej Woźniak <mlwozniak@wp.pl> - 2016-11-21 21:45 +0100
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-21 13:17 -0800
                Re: ds^2 and metric tensor Tom Roberts <tjroberts137@sbcglobal.net> - 2016-11-21 16:17 -0600
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-21 16:15 -0800
                Re: ds^2 and metric tensor mlwozniak@wp.pl - 2016-11-21 23:31 -0800
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-22 00:36 -0800
                Re: ds^2 and metric tensor mlwozniak@wp.pl - 2016-11-22 01:53 -0800
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-22 14:06 -0800
                Re: ds^2 and metric tensor alsor@interia.pl - 2016-11-22 14:31 -0800
                Re: ds^2 and metric tensor "David (Lord Kronos) Fuller" <fuller.david@hotmail.com> - 2016-11-22 14:42 -0800
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-22 15:41 -0800
                Re: ds^2 and metric tensor alsor@interia.pl - 2016-11-24 12:50 -0800
                Re: ds^2 and metric tensor mlwozniak@wp.pl - 2016-11-22 23:18 -0800
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-22 23:49 -0800
                Re: ds^2 and metric tensor mlwozniak@wp.pl - 2016-11-23 02:34 -0800
                Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-23 14:58 -0800
                Re: ds^2 and metric tensor mlwozniak@wp.pl - 2016-11-23 23:31 -0800
        Re: ds^2 and metric tensor alsor@interia.pl - 2016-11-20 08:17 -0800
          Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-20 20:45 -0800
            Re: ds^2 and metric tensor alsor@interia.pl - 2016-11-22 11:45 -0800
              Re: ds^2 and metric tensor JanPB <filmart@gmail.com> - 2016-11-23 16:38 -0800
                Re: ds^2 and metric tensor alsor@interia.pl - 2016-11-24 12:56 -0800
  Re: ds^2 and metric tensor alsor@interia.pl - 2016-11-19 09:15 -0800
    Re: ds^2 and metric tensor Norberto Haman <norberman@norberman.info> - 2016-11-19 17:25 +0000
      Re: ds^2 and metric tensor alsor@interia.pl - 2016-11-19 11:21 -0800

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